What is Graph Theory in Maths?
The concept of graph theory in maths plays a key role in mathematics and is widely applicable to real-life situations like networking, transport, and computer science, as well as exam problem-solving for board exams and competitive tests like JEE.
What Is Graph Theory in Maths?
A graph in mathematics is a collection of points, called vertices (or nodes), connected by lines, called edges. The study of these objects—their relationships, types, and applications—is called graph theory in maths. You’ll find this concept applied in areas such as network analysis, computer programming, and operations research.
Key Formula for Graph Theory in Maths
Here’s the standard formula used often in graph theory:
Degree Sum Formula: \( \sum \text{deg}(v) = 2 \times E \)
Where \( \text{deg}(v) \) is the degree of vertex \( v \) and \( E \) is the number of edges.
Euler’s Formula (for planar graphs): \( V - E + F = 2 \)
Where \( V \) = vertices, \( E \) = edges, \( F \) = faces.
Key Terminology Explained
| Term | Meaning |
|---|---|
| Vertex (Node) | A point in the graph where edges meet |
| Edge (Link) | A line connecting two vertices |
| Degree | The number of edges attached to a vertex |
| Directed Graph | Edges have direction (arrows); also called digraph |
| Undirected Graph | Edges have no direction (simple lines) |
| Complete Graph | Every pair of vertices is connected |
| Cycle | A path that starts and ends at the same vertex, with no repeats |
| Tree | A connected graph with no cycles |
Cross-Disciplinary Usage
Graph theory in maths is not only useful in Maths but also plays an important role in Physics, Computer Science, and even social sciences. Whether you are drawing networks, planning routes, or analysing data, these concepts are essential for advanced studies and competitive exams.
Types of Graphs with Examples
- Simple Graph: No loops, no multiple edges.
- Complete Graph: Each node connects to every other node.
- Directed Graph: Edges have arrows showing direction.
- Weighted Graph: Each edge has a value (weight) assigned.
- Tree: A kind where there's exactly one path between any two vertices.
- Cyclic Graph: Contains at least one cycle.
For visual learners, you can think of Facebook's friend network as an undirected graph, while Twitter followership is modeled as a directed graph. Explore more real-case diagrams here.
Step-by-Step Illustration: Sample Problem
Question: In a simple graph with 4 vertices and 5 edges, what is the sum of degrees of all vertices?
1. The degree sum formula says: Sum of degrees = \( 2 \times \) (Number of edges)2. Substitute: \( 2 \times 5 = 10 \)
3. Final Answer: The sum of degrees = 10.
Get more worked examples and exam-level practice here.
Speed Trick or Vedic Shortcut
A quick tip: To identify if a given graph is Eulerian (a path/circuit covering every edge exactly once), check if all vertices have even degrees. If so, the graph has an Eulerian circuit! These checks save precious minutes in board and JEE exams.
Learn more about Euler's Theorem in graph theory for exam shortcuts.
Applications in Real Life
- Network design (Internet, social media connections)
- Transport and route planning
- Project scheduling (PERT networks, Gantt charts)
- Biology (food webs, neural networks)
- Computer Science (search algorithms, database design)
Try These Yourself
- Draw a graph with 3 vertices and 2 edges.
- Is a tree always a connected graph? Why?
- List all types of graphs you know and give 1 example of each.
- Check if the following degree sequence is possible: 2, 2, 2, 2.
Frequent Errors and Misunderstandings
- Confusing between directed and undirected edges.
- Miscounting degrees (remember: loops count as two in degree calculation).
- Not checking if a path repeats edges when solving for Euler circuits.
Relation to Other Concepts
The idea of graph theory in maths connects closely with discrete mathematics, set theory, and logic. Mastering graphs will boost your problem-solving in many advanced maths topics, including algorithms and computer networks.
Classroom Tip
A great way to remember: "Vertices are dots, edges are lines. A cycle means you can start and end at the same point without retracing steps." Vedantu’s teachers often use color-coded sketches to help visualize graph questions for quick understanding.
We explored graph theory in maths—from definition, formula, examples, speed checks, and applications to connections with other topics. Practice regularly with Vedantu’s quizzes and worksheets to become confident and quick at solving graph-based problems in any exam or real-life scenario.
Recommended next steps:
- Types of Graphs in Maths
- Euler’s Theorem Explained
- Set Theory Symbols (for graph notation)
- Discrete Mathematics Overview
FAQs on Graph Theory: Definition, Concepts & Problem Solving
1. What is graph theory in maths?
Graph theory in maths is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices (or nodes) connected by edges. It's a fundamental part of discrete mathematics with wide applications in computer science, network analysis, and various other fields. Graph theory helps us understand and solve problems related to networks, relationships, and optimization.
2. What are the different types of graphs?
There are many types of graphs, categorized by their properties:
- Directed Graphs: Edges have a direction (one-way).
- Undirected Graphs: Edges have no direction (two-way).
- Weighted Graphs: Edges have assigned weights (representing cost, distance, etc.).
- Simple Graphs: No loops or multiple edges between the same vertices.
- Complete Graphs: Every pair of distinct vertices is connected by a unique edge.
- Trees: Connected graphs without cycles, often used in data structures.
- Bipartite Graphs: Vertices can be divided into two sets, with edges only connecting vertices from different sets.
3. What are vertices and edges in graph theory?
In graph theory, vertices (also called nodes or points) represent the objects in a network. Edges (or lines or arcs) represent the connections or relationships between these objects. The number of edges connected to a vertex is called its degree.
4. What are some applications of graph theory?
Graph theory finds applications in diverse fields:
- Computer Science: Data structures, algorithms, network routing, database design.
- Network Analysis: Social networks, transportation networks, communication networks.
- Operations Research: Scheduling, optimization, assignment problems.
- Chemistry: Molecular structures, chemical reactions.
- Biology: Modeling biological networks, analyzing genetic data.
5. What is Euler's theorem in graph theory?
Euler's theorem states that a connected graph has an Eulerian circuit (a path that visits every edge exactly once and returns to the starting vertex) if and only if every vertex has an even degree. If it has exactly two vertices of odd degree, it has an Eulerian path (a path that visits every edge exactly once).
6. What is a tree in graph theory?
A tree is a connected undirected graph with no cycles (no closed paths). It has a unique path between any two vertices. Trees are fundamental data structures in computer science.
7. What is the difference between a directed and an undirected graph?
In a directed graph, edges have a direction, indicating a one-way relationship between vertices. In an undirected graph, edges have no direction, representing a two-way relationship.
8. What is graph coloring?
Graph coloring is the assignment of colors to the vertices of a graph such that no two adjacent vertices have the same color. The minimum number of colors required is called the chromatic number. It has applications in scheduling and resource allocation.
9. What is a complete graph?
A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete graph with n vertices is denoted as Kn.
10. What are some important graph algorithms?
Several important algorithms operate on graphs:
- Breadth-First Search (BFS): Explores a graph level by level.
- Depth-First Search (DFS): Explores a graph by going as deep as possible along each branch before backtracking.
- Dijkstra's algorithm: Finds the shortest path between two vertices in a weighted graph.
- Prim's algorithm and Kruskal's algorithm: Find the minimum spanning tree of a weighted graph.
11. How is graph theory used in social network analysis?
Graph theory is essential for analyzing social networks. Individuals are represented as vertices, and relationships (friendships, connections, etc.) as edges. Algorithms can identify influential individuals, community structures, and information diffusion patterns within the network.
